1 Analytical solution for coupled non-Fickian diffusion-thermoelasticity and thermoelastic wave propagation analysis Seyed Amin Hosseini 1 , Seyed Mahmoud Hosseini 2 , Mohammad Hossein Abolbashari 3* 1 Mechanical Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, PO Box: 91775-1111, Mashhad, Iran, Email: [email protected]2 Industrial Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, PO Box: 91775-1111, Mashhad, Iran, Email: [email protected]3* Department of Mechanical Engineering, Lean Production Engineering Research Center, Ferdowsi University of Mashhad, PO Box: 91775-1111, Mashhad, Iran [email protected]Abstract The time history analysis and propagation of molar concentration, temperature and displacement waves are studied in details using an analytical method. The method is applied to coupled non-Fickian diffusion- thermoelasticity analysis of a strip. The governing equations are derived using non-Fickian theory of diffusion and classic theories for coupled thermoelasticity. Molar concentration and thermoelastic wave propagations are considered to be of finite speed. The governing equations are first transferred to the frequency domain using Laplace transform technique. The unknown parameters are then obtained in analytical forms proposed by the presented method. By employing the Talbot technique, the unknown parameters are eventually determined in time domain. It can be concluded that the presented analytical method has a high capability for dynamic and transient analysis of coupled diffusion-thermoelasticity problems. The wave fronts in displacement, temperature and molar concentration fields can be tracked at various time instants employing the presented analytical method. Keywords Non-Fickian diffusion; wave propagation; molar concentration; temperature; analytical method; coupled problems. Nomenclature A Temperature constant 0 T Reference temperature A n (s),B n (s),D n (s) Unknown coefficients i u Components of displacement vector c Mass concentration x Position c ˆ Specific heat , ij Mechanical diffusion coefficient 0 c Reference concentration t Coefficient of linear thermal expansion 1 c Shock concentration 1 1 , ij Mechanical-thermo coefficient ijkl C The elastic constants Chemical potential constant 0 D Diffusion coefficient Constant coefficient
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Analytical solution for coupled non-Fickian diffusion-thermoelasticity and
thermoelastic wave propagation analysis
Seyed Amin Hosseini1, Seyed Mahmoud Hosseini
2, Mohammad Hossein Abolbashari
3*
1 Mechanical Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, PO
Box: 91775-1111, Mashhad, Iran, Email: [email protected] 2 Industrial Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, PO
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Seyed Amin Hosseini received his Ph.D. in mechanical engineering from Ferdowsi University of Mashhad, Iran in 2017. He is currently an assistant professor in the mechanical engineering department at Chabahar Maritime University. His research interests include computational mechanics, finite element method, fracture, and thermal stresses. He has published 10 journal and conference paper.
Seyed Mahmoud Hosseini received his Ph.D. degree in Mechanical Engineering from Amirkabir University of Technology (Tehran Polytechnic), Iran in 2007. He started his work at Ferdowsi University of Mashhad in 2009. He is currently an associate professor in Ferdowsi University of Mashhad, Iran. His main focus of research is within dynamic analysis and wave propagation in solids, coupled and uncoupled thermoelasticity analysis, and stochastic and reliability analysis of stress field using analytical and numerical methods (including FEM, MLPG and GFD methods). He has more than 80 peer-reviewed journal and conference publications. Mohammad Hossein Abolbashari is currently Professor of Mechanical Engineering at Ferdowsi University of Mashhad, Iran. He received his Ph.D. from the University of Saskatchewan, Canada, in 1995. He is director of the “Lean Production Engineering Research Center. He has recently joined the industrial engineering department of Ferdowsi University of Mashhad as an honored member. His research interests include structural optimization, computational mechanics, finite elements methods and time dependent materials. He has authored/coauthored more than 112 refereed journal and conference publications. He serves as the editorial board of two refereed journals namely: “Journal of Applied and Computational Sciences in Mechanics” and “Journal of Solid and Fluid Mechanics“ which are currently published in Farsi (Persian). He is a fellow of the Iranian Society of Mechanical Engineers (ISME) and the Iranian Organization for Engineering Order of Building.
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Table 1. Material specifications and other parameters of the problem
Figure 1: Schematic of the boundary conditions for the strip
Figure 2: The comparison between obtained results and those of ref [34] for molar concentration along the x
direction.
Figure 3: The dynamic behavior of the displacement at different positions in time domain
Figure 4: The dynamic behavior of the molar concentration at different positions in time domain
Figure 5: The dynamic behavior of the temperature at different positions in time domain
Figure 6: The wave fronts of molar concentration wave propagation along the “ x ” direction for various time
instants
Figure 7: The wave fronts of displacement wave propagation along the “ x ” direction for various time instants
Figure 8: The wave fronts of temperature wave propagation along the “ x ” direction for various time instants
Figure 9: The comparison between classical and non-classical form of concentration along the “ x ” direction
Figure 10: The effect of relaxation time on concentration along the “ x ” direction
Table 1. Material specifications and other parameters of the problem
)(10326.12
9
m
N )(10884.0
2
9
m
N )(2000
3m
kg )(10000
2
0s
mD
).
(58322
4
mol
mN )
.(87082.1mol
mN )(10086.3 1
0 s )1
(1078.1 5
Kt
).
(1000sK
Nk
)
.(8.1
2 Km
NA
)
.
.(0001.01
Kmol
mN
).
.(5.24ˆˆ
2 Km
sNc
)(9354031
m
molc )(3000 KT t 231 )(4501 K
13
Figure 1: Schematic of the boundary conditions for the strip
Figure 2: The comparison between obtained results and those of ref [34] for molar concentration along the x
direction.
u=0
0
c=0
)(
)(
0
1
1
tHcc
tH
1.5 m
x
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Figure 3: The dynamic behavior of the displacement at different positions in time domain
Figure 4: The dynamic behavior of the molar concentration at different positions in time domain
Figure 5: The dynamic behavior of the temperature at different positions in time domain
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Figure 6: The wave fronts of molar concentration wave propagation along the “ x ” direction for various time
instants
Figure 7: The wave fronts of displacement wave propagation along the “ x ” direction for various time instants
Figure 8: The wave fronts of temperature wave propagation along the “ x ” direction for various time instants
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Figure 9: The comparison between classical and non-classical form of concentration along the “ x ” direction
Figure 10: The effect of relaxation time on concentration along the “ x ” direction